5 Recall [[the recent homework assignment|/exercises/assignment12]] where you solved the same-fringe problem with a `make_fringe_enumerator` function, or in the Scheme version using streams instead of zippers, with a `lazy-flatten` function.

7 The technique illustrated in those solutions is a powerful and important one. It's an example of what's sometimes called **cooperative threading**. A "thread" is a subprogram that the main computation spawns off. Threads are called "cooperative" when the code of the main computation and the thread fixes when control passes back and forth between them. (When the code doesn't control this---for example, it's determined by the operating system or the hardware in ways that the programmer can't predict---that's called "preemptive threading.") Cooperative threads are also sometimes called *coroutines* or *generators*.

39 The way we built cooperative threads using `make_fringe_enumerator` crucially relied on two heavyweight tools. First, it relied on our having a data structure (the tree zipper) capable of being a static snapshot of where we left off in the tree whose fringe we're enumerating. Second, it either required us to manually save and restore the thread's snapshotted state (a tree zipper); or else we had to use a mutable reference cell to save and restore that state for us. Using the saved state, the next invocation of the `next_leaf` function could start up again where the previous invocation left off.

41 It's possible to build cooperative threads without using those tools, however. Already our [[solution using streams|/exercises/assignment12#streams2]] uses neither zippers nor any mutation. Instead it saves the thread's state in the code of explicitly-created thunks, and resumes the thread by forcing the thunk.

81 We're going to think about the underlying principles to this execution pattern, and instead learn how to implement it from scratch---without necessarily having zippers or dedicated native syntax to rely on.

88 While writing OCaml code, you've probably come across errors. In fact, you've probably come across errors of several sorts. One sort of error comes about when you've got syntax errors and the OCaml interpreter isn't even able to parse your code. A second sort of error is type errors, as in:

96 Type errors are also detected and reported before OCaml attempts to execute or evaluate your code. But you may also have encountered a third kind of error, that arises while your program is running. For example:

103 These "Exceptions" are **run-time errors**. OCaml will automatically detect some of them, like when you attempt to divide by zero. Other exceptions are manually *raised* by code. For instance, here is the standard implementation of `List.nth`:

113 (The Juli8 version of `List.nth` only differs in sometimes raising a different error.) Notice the two clauses `invalid_arg "List.nth"` and `failwith "nth"`. These are two helper functions which are shorthand for:

139 the effect is for the program to immediately stop. That's not exactly true. You can also programmatically arrange to *catch* errors, without the program necessarily stopping. In OCaml we do that with a `try ... with PATTERN -> ...` construct, analogous to the `match ... with PATTERN -> ...` construct. (In OCaml 4.02 and higher, there is also a more inclusive construct that combines these, `match ... with PATTERN -> ... | exception PATTERN -> ...`.)

157 Notice what happens here. If we call `foo 1`, then the code between `try` and `with` evaluates to `110`, with no exceptions being raised. That then is what the entire `try ... with ...` block evaluates to; and so too what `foo 1` evaluates to. If we call `foo 2`, then the code between `try` and `with` raises an exception `Failure "two"`. The pattern in the `with` clause matches that exception, so we get instead `20`. If we call `foo 3`, we again raise an exception. This exception isn't matched by the `with` block, so it percolates up to the top of the program, and then the program immediately stops.

187 Of course, it's possible to handle errors in other ways too. There's no reason why the implementation of `List.nth` *had* to raise an exception. They might instead have returned `Some a` when the list had an nth member `a`, and `None` when it does not. But it's pedagogically useful for us to think about the exception-raising pattern now.

189 When an exception is raised, it percolates up through the code that called it, until it finds a surrounding `try ... with ...` that matches it. That might not be the first `try ... with ...` that it encounters. For example:

211 Here we call `foo bar 0`, and `foo` in turn calls `bar 0`, and `bar` raises the exception. Since there's no matching `try ... with ...` block in `bar`, we percolate back up the history of who called that function, and we find a matching `try ... with ...` block in `foo`. This catches the error and so then the `try ... with ...` block in `foo` (the code that called `bar` in the first place) will evaluate to `20`.

213 OK, now this exception-handling apparatus does exemplify the second execution pattern we want to focus on. But it may bring it into clearer focus if we **simplify the pattern** even more. Imagine we could write code like this instead:

223 then if we called `foo 1`, we'd get the result `110`. If we called `foo 2`, on the other hand, we'd get `20` (note, not `120`). This exemplifies the same interesting "jump out of this part of the code" behavior that the `try ... raise ... with ...` code does, but without the details of matching which exception was raised, and handling the exception to produce a new result.

225 Many programming languages have this simplified exceution pattern, either instead of or alongside a `try ... with ...`-like pattern. In Lua and many other languages, `abort` is instead called `return`. In Lua, the preceding example would be written:

248 In both of these patterns --- coroutines and exceptions/aborts --- we need to have some way to take a snapshot of where we are in the evaluation of a complex piece of code, so that we might later resume execution at that point. In the coroutine example, the two threads need to have a snapshot of where they were in the enumeration of their tree's leaves. In the abort example, we need to have a snapshot of where to pick up again if some embedded piece of code aborts. Sometimes we might distill that snapshot into a data structure like a zipper. But we might not always know how to do so; and learning how to think about these snapshots without the help of zippers will help us see patterns and similarities we might otherwise miss.

295 That function is our "snapshot". Normally what happens is that code *inside* the box delivers up a value, and that value gets supplied as an argument to the snapshot-function just described. That is, our code is essentially working like this:

309 Well, that's when we use the snapshot code in an unusual way. If we encounter an `abort 20`, we should abandon the code we're currently executing, and instead just supply `20` to the snapshot we saved when we entered the box. That is, something like this:

321 Except that isn't quite right, yet---in this fragment, after the `snapshot 20` code is finished, we'd pick up again inside `let foo_applied_to_x = (...) + 100 in snapshot foo_applied_to_x`. That's not what we want. We don't want to pick up again there. We want instead to do this:

364 You can think of them as functions that represent "how the rest of the computation proposes to continue." Except that, once we're able to get our hands on those functions, we can do exotic and unwholesome things with them. Like use them to suspend and resume a thread. Or to abort from deep inside a sub-computation: one function might pass the command to abort *it* to a subfunction, so that the subfunction has the power to jump directly to the outside caller. Or a function might *return* its continuation function to the outside caller, giving *the outside caller* the ability to "abort" the function (the function that has already returned its value---so what should happen then?) Or we may call the same continuation function *multiple times* (what should happen then?). All of these weird and wonderful possibilities await us.

366 The key idea behind working with continuations is that we're *inverting control*. In the fragment above, the code `(if x = 1 then ... else snapshot 20) + 100`---which is written as if it were to supply a value to the outside context that we snapshotted---itself *makes non-trivial use of* that snapshot. So it has to be able to refer to that snapshot; the snapshot has to somehow be available to our inside-the-box code as an *argument* or bound variable. That is: the code that is *written* like it's supplying an argument to the outside context is instead *getting that context as its own argument*. He who is written as value-supplying slave is instead become the outer context's master.

368 In fact you've already seen this several times this semester---recall how in our implementation of pairs in the untyped lambda-calculus, the handler who wanted to use the pair's components had *in the first place to be supplied to the pair as an argument*. So the exotica from the end of the seminar was already on the scene in some of our earliest steps. Recall also what we did with our [[abortable list traversals|/topics/week12_abortable_traversals]].

370 This inversion of control should also remind you of Montague's treatment of determiner phrases in ["The Proper Treatment of Quantification in Ordinary English"](http://www.blackwellpublishing.com/content/BPL_Images/Content_store/Sample_chapter/0631215417%5CPortner.pdf) (PTQ).

372 A naive semantics for atomic sentences will say the subject term is of type `e`, and the predicate of type `e -> t`, and that the subject provides an argument to the function expressed by the predicate.

374 Monatague proposed we instead take the subject term to be of type `(e -> t) -> t`, and that now it'd be the predicate (still of type `e -> t`) that provides an argument to the function expressed by the subject.

376 If all the subject did then was supply an `e` to the `e -> t` it receives as an argument, we wouldn't have gained anything we weren't already able to do. But of course, there are other things the subject can do with the `e -> t` it receives as an argument. For instance, it can check whether anything in the domain satisfies that `e -> t`; or whether most things do; and so on.

380 Continuations come in many varieties. There are **undelimited continuations**, expressed in Scheme via `(call/cc (lambda (k) ...))` or the shorthand `(let/cc k ...)`. (`call/cc` is itself shorthand for `call-with-current-continuation`.) These capture "the entire rest of the computation." There are also **delimited continuations**, expressed in Scheme via `(reset ... (shift k ...) ...)` or `(prompt ... (control k ...) ...)` or any of several other operations. There are subtle differences between those that we won't be exploring in the seminar. Ken Shan has done terrific work exploring the relations of these operations to each other.

407 There are also different kinds of "syntactic sugar" we can use to hide the continuation plumbing. Of course we'll be talking about how to manipulate continuations **with a Continuation monad.** We'll also talk about a style of working with continuations where they're **mostly implicit**, but special syntax allows us to distill the implicit continuation into a first-class value (the `k` in `(let/cc k ...)` and `(shift k ...)`. For reference, here's how the preceding code looks, using Scheme's `abort` or `shift` operators:

460 Various of the tools we've been introducing over the past weeks are inter-related. We saw coroutines implemented first with zippers; here we've talked in the abstract about their being implemented with continuations. Oleg says that "Zipper can be viewed as a delimited continuation reified as a data structure." Ken expresses the same idea in terms of a zipper being a "defunctionalized" continuation---that is, take something implemented as a function (a continuation) and implement the same thing as an inert data structure (a zipper).

462 Mutation, delimited continuations, and monads can also be defined in terms of each other in various ways. We find these connections fascinating but the seminar won't be able to explore them very far.

464 We recommend reading [the Yet Another Haskell Tutorial on Continuation Passing Style](http://en.wikibooks.org/wiki/Haskell/YAHT/Type_basics#Continuation_Passing_Style)---though the target language is Haskell, this discussion is especially close to material we're discussing in the seminar.